1) is Fleet Street far?
Could I also say: far away? which one would you prefer?
It is correct. I would tend to add "away" - but yours is correct.
2)No, it isn't. It is NEAR /NEAR HERE/ NEARBY.
Are all these alternatives ok?
"Near" cries out for a something to be added. It is near the Palace. "Near here" is fine. "Nearby" works fine, as well, especially if you are not there when you are saying it. (Example, I want to visit the Castle. Is Fleet Street far from there? No, it's nearby.)
3) Go straight on along Oxford Street
Could I say : Go ahead along Oxford street?
Both are all right; I think the first is more natural.
4) Turn right at the second traffic lights, then take the first turning on the left. That is Fleet Street
Perfect. Although we would probably say "first turn" rather than "turning."
5) is there a post office in Fleet Street?
Only if Fleet Street is a district; I think it's not, but is rather just a street in the normal sense. In that case, you would say "on" Fleet Street. If it were a district, like Mayfair, you would say, "in Mayfair."
6) It's NEAR THERE/NEARBY/ IN THE NEARBY
The first and second are all right; not the third.
Are the alternatives ok?
7) Go along the road for about 50 metres, along the pavement
Fine. Especially if there are options or forks in the road where one fork is unpaved. I might say "Follow the road for ..." instead of "Go along..."
Remember I am on my moped. Is number 7 ok? and If I was on foot?